When \({\mathcal {M}}\)  is a smooth compact manifold with no boundary, \(\mu ,\nu \in {\mathcal {P}}({\mathcal {M}})\) , and \(\mu \)  has density (i.e., is absolutely continuous w.r.t. the volume measure of \({\mathcal {M}}\) ), Theorem 9 of [1]} shows that there is a unique (up-to \(\mu \) -zero sets) transport map \(t:{\mathcal {M}}\rightarrow {\mathcal {M}}\)  that pushes \(\mu \)  to \(\nu \) , i.e., \(s_\#\mu =\nu \) , and minimizes the transport cost
\(C(s)=\int _{\mathcal {M}}c(x,s(x)) d\mu (x).\) 
